<s>
False	B-Algorithm
diffusion	I-Algorithm
is	O
a	O
type	O
of	O
error	O
observed	O
when	O
the	O
upwind	B-Algorithm
scheme	I-Algorithm
is	O
used	O
to	O
approximate	O
the	O
convection	O
term	O
in	O
convection	O
–	O
diffusion	O
equations	O
.	O
</s>
<s>
The	O
more	O
accurate	O
central	O
difference	O
scheme	O
can	O
be	O
used	O
for	O
the	O
convection	O
term	O
,	O
but	O
for	O
grids	O
with	O
cell	O
Peclet	O
number	O
more	O
than	O
2	O
,	O
the	O
central	O
difference	O
scheme	O
is	O
unstable	O
and	O
the	O
simpler	O
upwind	B-Algorithm
scheme	I-Algorithm
is	O
often	O
used	O
.	O
</s>
<s>
The	O
resulting	O
error	O
from	O
the	O
upwind	B-Algorithm
differencing	I-Algorithm
scheme	I-Algorithm
has	O
a	O
diffusion-like	O
appearance	O
in	O
two	O
-	O
or	O
three-dimensional	O
co-ordinate	O
systems	O
and	O
is	O
referred	O
as	O
"	O
false	B-Algorithm
diffusion	I-Algorithm
"	O
.	O
</s>
<s>
False-diffusion	O
errors	O
in	O
numerical	B-General_Concept
solutions	I-General_Concept
of	O
convection-diffusion	O
problems	O
,	O
in	O
two	O
-	O
and	O
three-dimensions	O
,	O
arise	O
from	O
the	O
numerical	B-General_Concept
approximations	I-General_Concept
of	O
the	O
convection	O
term	O
in	O
the	O
conservation	O
equations	O
.	O
</s>
<s>
Over	O
the	O
past	O
20	O
years	O
many	O
numerical	B-General_Concept
techniques	O
have	O
been	O
developed	O
to	O
solve	O
convection-diffusion	O
equations	O
and	O
none	O
are	O
problem-free	O
,	O
but	O
false	B-Algorithm
diffusion	I-Algorithm
is	O
one	O
of	O
the	O
most	O
serious	O
problems	O
and	O
a	O
major	O
topic	O
of	O
controversy	O
and	O
confusion	O
among	O
numerical	B-General_Concept
analysts	I-General_Concept
.	O
</s>
<s>
False	B-Algorithm
diffusion	I-Algorithm
is	O
defined	O
as	O
an	O
error	O
having	O
a	O
diffusion-like	O
appearance	O
,	O
obtained	O
when	O
the	O
upwind	B-Algorithm
scheme	I-Algorithm
is	O
used	O
in	O
multidimensional	O
cases	O
to	O
solve	O
for	O
the	O
distribution	O
of	O
transported	O
properties	O
flowing	O
non-orthogonally	O
to	O
one	O
or	O
more	O
of	O
the	O
system	O
's	O
major	O
axes	O
.	O
</s>
<s>
There	O
will	O
be	O
no	O
diffusion	O
across	O
the	O
diagonal	O
XX	O
but	O
,	O
when	O
the	O
upwind	B-Algorithm
scheme	I-Algorithm
is	O
applied	O
the	O
results	O
are	O
similar	O
to	O
case	O
(	O
i	O
)	O
where	O
actual	O
diffusion	O
is	O
occurring	O
.	O
</s>
<s>
This	O
error	O
is	O
known	O
as	O
false	B-Algorithm
diffusion	I-Algorithm
.	O
</s>
<s>
In	O
early	O
approaches	O
,	O
derivatives	B-Algorithm
in	O
the	O
differential	O
form	O
of	O
the	O
governing	O
transport	O
equation	O
were	O
replaced	O
by	O
finite	O
difference	O
approximations	O
,	O
usually	O
central	O
differencing	O
approximations	O
with	O
second	O
order	O
accuracy	O
.	O
</s>
<s>
It	O
was	O
recognized	O
independently	O
by	O
several	O
investigators	O
that	O
the	O
less	O
expensive	O
but	O
only	O
first	O
order	O
accurate	O
upwind	B-Algorithm
scheme	I-Algorithm
can	O
be	O
employed	O
but	O
that	O
this	O
scheme	O
produces	O
results	O
with	O
false	B-Algorithm
diffusion	I-Algorithm
for	O
multidimensional	O
cases	O
.	O
</s>
<s>
Many	O
new	O
schemes	O
have	O
been	O
developed	O
to	O
counter	O
false	B-Algorithm
diffusion	I-Algorithm
but	O
a	O
reliable	O
,	O
accurate	O
and	O
economical	O
discretisation	O
scheme	O
is	O
still	O
unavailable	O
.	O
</s>
<s>
False	B-Algorithm
diffusion	I-Algorithm
with	O
the	O
upwind	B-Algorithm
scheme	I-Algorithm
is	O
reduced	O
by	O
increasing	O
the	O
mesh	O
density	O
.	O
</s>
<s>
In	O
the	O
results	O
of	O
figure	O
3	O
and	O
4	O
the	O
false	B-Algorithm
diffusion	I-Algorithm
error	O
is	O
lowest	O
in	O
figure	O
4(b )	O
with	O
finer	O
mesh	O
size	O
.	O
</s>
<s>
False	B-Algorithm
diffusion	I-Algorithm
error	O
also	O
can	O
be	O
reduced	O
by	O
using	O
schemes	O
such	O
as	O
the	O
power	O
law	O
scheme	O
,	O
QUICK	O
scheme	O
,	O
exponential	O
scheme	O
,	O
and	O
SUCCA	O
,	O
and	O
others	O
.	O
</s>
<s>
False	B-Algorithm
diffusion	I-Algorithm
with	O
the	O
simple	O
upwind	B-Algorithm
scheme	I-Algorithm
occurs	O
because	O
the	O
scheme	O
does	O
not	O
take	O
into	O
account	O
grid/flow	O
direction	O
inclination	O
.	O
</s>
<s>
False	B-Algorithm
diffusion	I-Algorithm
is	O
absent	O
when	O
the	O
resultant	O
flow	O
is	O
aligned	O
with	O
either	O
of	O
the	O
sets	O
of	O
grid	O
lines	O
and	O
is	O
greatest	O
when	O
the	O
flow	O
direction	O
is	O
45˚	O
to	O
the	O
grid	O
lines	O
.	O
</s>
<s>
The	O
subscript	O
fc	O
is	O
a	O
reminder	O
that	O
this	O
is	O
a	O
false	B-Algorithm
diffusion	I-Algorithm
arising	O
from	O
the	O
estimate	O
of	O
the	O
convected	O
flux	O
at	O
the	O
instant	O
using	O
UAC	O
.	O
</s>
<s>
This	O
formulation	O
satisfies	O
all	O
the	O
criteria	O
of	O
convergence	B-Algorithm
and	O
stability	O
.	O
</s>
<s>
6	O
,	O
as	O
mesh	O
is	O
refined	O
,	O
the	O
upwind	B-Algorithm
scheme	I-Algorithm
gives	O
more	O
accurate	O
results	O
but	O
SUCCA	O
offers	O
a	O
nearly	O
exact	O
solution	O
and	O
is	O
more	O
useful	O
in	O
avoiding	O
multidimensional	O
false	B-Algorithm
diffusion	I-Algorithm
errors	O
.	O
</s>
